Answer
$\left\{0.3649\text{ radians}, 3.5065\text{ radians}, 1.2059\text{ radians}, 4.3475\text{ radians}\right\}$
Work Step by Step
Multiply $2$ t both sides of the equation:
$2\sin{x}\cos{x} = \frac{2}{3}$
RECALL:
$\sin{(2x)} = 2\sin{x}\cos{x}$
Use the identity above to obtain:
$2\sin{x}\cos{x}=\frac{2}{3}
\\\sin{(2x)} = \frac{2}{3}
\\2x=\sin^{-1}{(\frac{2}{3})}$
Use a scientific calculator's inverse sine function to obtain:
$2x=0.7297276562$ radians or $2x=2.411864997$ \radians
Since the period of $sin{x}$ is $2\pi$, add $2\pi$ to each solution above to obtain two more solutions:
$2x= 7.012912963$ radians or $2x=8.695050305$ radians
Divide both sides of each equation by $2$, then round off the answer to four decimal places to obtain:
$x\approx 0.3649, 3.5065, 1.2059, 4.3475$ radians
